Simplify the following expression and state the condition under which the simplification is valid. You can assume that $z \neq 0$. $t = \dfrac{6z}{45z - 36} \div \dfrac{3z}{10z - 8} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{6z}{45z - 36} \times \dfrac{10z - 8}{3z} $ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ 6z \times (10z - 8) } { (45z - 36) \times 3z } $ $ t = \dfrac {6z \times 2(5z - 4)} {3z \times 9(5z - 4)} $ $ t = \dfrac{12z(5z - 4)}{27z(5z - 4)} $ We can cancel the $5z - 4$ so long as $5z - 4 \neq 0$ Therefore $z \neq \dfrac{4}{5}$ $t = \dfrac{12z \cancel{(5z - 4})}{27z \cancel{(5z - 4)}} = \dfrac{12z}{27z} = \dfrac{4}{9} $